1. Field of Invention
The present invention relates to physical data decomposition. More particularly, the present invention relates to decomposing and analyzing spatial physical data.
2. Description of Related Art
The combination of Empirical Mode Decomposition (EMD) with the Hilbert Spectral Analysis (HSA) designated as the Hilbert-Huang Transform (HHT), in Patents number 1 to 5 by the National Aeronautics and Space Administration (NASA), has provided an alternative paradigm in time-frequency analysis. The Hilbert Transform is well known and has been widely used in the signal processing field since the 1940s (Gabor 1946). However, the Hilbert Transform has many drawbacks (Bedrosian 1963, Nuttall 1966) when applied to instantaneous frequency computation. The most serious drawback is that the derived instantaneous frequency of a signal could lose its physical meaning when the signal is not a mono-component or AM/FM separable oscillatory function (Huang et al. 1998). The EMD, at its very beginning (Huang, et al. 1996, 1998, and 1999), was developed to overcome this drawback so that the data can be examined in a physically meaningful time-frequency-amplitude space. It has been widely accepted that the EMD, with its new improvements (Huang et al. 2003, Wu and Huang 2004, Wu and Huang 2005a, 2005b, Wu et al. 2007, Wu and Huang 2008a, Huang et al. 2008), has become a powerful tool in both signal processing and scientific data analysis (Huang and Attoh-Okine 2005 and Huang and Shen 2005, Huang and Wu 2008b).
Contrary to almost all the previous decomposition methods, EMD is empirical, intuitive, direct, and adaptive, without pre-determined basis functions. The decomposition is designed to seek the different simple intrinsic modes of oscillations in any data based on local time scales. A simple oscillatory mode is called intrinsic mode function (IMF) which satisfies: (a) in the whole data set, the number of extrema (maxima value or minima value) and the number of zero-crossings must either equal or differ at most by one; and (b) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.
The EMD is implemented through a sifting process that uses only local extrema. From any data rj-1, say, the procedure is as follows: 1) identify all the local extrema and connect all local maxima (minima) with a cubic spline as the upper (lower) envelope; 2) obtain the first component h by taking the difference between the data and the local mean of the two envelopes; and 3) treat h as the data and repeat steps 1 and 2 until the envelopes are symmetric with respect to zero mean under certain criterion. The final h is designated as cj. A complete sifting process stops when the residue, rn, becomes a monotonic function or a function with only one extrema from which no more IMF can be extracted. In short, the EMD is an adaptive method that decompose data x(t) in terms of IMFs cj and a residual component rn, i.e.,
                              x          ⁡                      (            t            )                          =                                            ∑                              j                =                1                            n                        ⁢                          c              j                                +                      r            n                                              (        1        )            
In Eq. (1), the residual component rn could be a constant, a monotonic function, or a function that contains only a single extrema, from which no more IMF can be extracted. In this way, the decomposition method is adaptive, and, therefore, highly efficient. As the decomposition is based on the local characteristics of the data, it is applicable to nonlinear and nonstationary processes.
Although EMD is a simple decomposition method, it has many wonderful characteristics that other decomposition methods lack. Flandrin et al. (2004 and 2005), Flandrin and Gonçalves (2004) studied the Fourier spectra of IMFs of fractional Gaussian noise, which are widely used in the signal processing community and financial data simulation. They found that the spectra of all IMFs except the first one of any fractional Gaussian noise collapse to a single shape along the axis of the logarithm of frequency (or period) with appropriate amplitude scaling of the spectra. The center frequencies (periods) of the spectra of the neighboring IMFs are approximately halved (and hence doubled); therefore, the EMD is essentially a dyadic filter bank. Flandrin et al. (2005) also demonstrated that EMD behaves like a cubic spline wavelet when it is applied to Delta functions. Independently, Wu and Huang (2004, 2005a) found the same result for white noise (which is a special case of fractional Gaussian noise). In addition to that, Wu and Huang (2004, 2005a) argued using the central limit theorem that each IMF of Gaussian noise is approximately Gaussian distributed, and therefore, the energy of each IMF must be a X2 distribution. From the characteristics they obtained, Wu and Huang (2004, 2005) further derived the expected energy distribution of IMFs of white noise. By determining the number of degrees of freedom of the X2 distribution for each IMF of noise, they derived the analytic form of the spread function of the energy of IMF. From these results, one would be able to discriminate an IMF of data containing signals from that of only white noise with any arbitrary statistical significance level. They verified their analytic results with those from the Monte Carlo test and found consistency.
The powerfulness of EMD has stimulated the development of two-dimensional EMD (or bi-dimensional EMD, BEMD, as will be referred later). By far, many researchers have explored the possibility of extending EMD for multi-dimensional spatial-temporal data analysis and for spatially two-dimensional image analysis. One method is to treat a two-dimensional image as a collection of one-dimensional slices and then decompose each slice using one-dimensional EMD. This is a pseudo-two-dimensional EMD. The first attempt of such type was initiated by Huang (2001, in Patent number 2). This method was later used by Long (2005) on wave data and produced excellent patterns and statistics of surface ripple riding on underlying long waves. In general, such an approach seems to work well in some cases of dealing with temporal-spatial data when a dominant direction could be identified clearly. However, in most of cases of pseudo-BEMD, the spatial structure is essentially determined by textual scales. If spatial structures of different textual scales are easily distinguishable, with clear directionality and without intermittency, this approach would be appropriate. If it is not, the applicability of this approach is significantly reduced. The main shortcoming of this approach is the inter-slice discontinuity due to EMD being sensitive to small data perturbation, intermittency and highly variable directionality.
The second type of effort is to directly transplant the idea and algorithm behind the EMD for image decomposition. As it has been introduced early, while EMD is a one-dimensional data decomposition method, its essential step of fitting extrema of one-dimensional data with upper and lower curves (envelopes) using a cubic spline or low order polynomials is applicable straightforwardly to two dimensional images, with fitting surfaces replacing fitting curves. Currently, there are several versions of genuine two-dimensional EMD, each containing a fitting surface determined by its own method. Nunes et al. (2003a, b, and 2005) used a radial basis function for surface interpretation, and the Riesz transform rather than the Hilbert transform for computing the local wave number. Linderhed (2005) used the thin-plate spline for surface interpretation to develop two-dimensional EMD data for an image compression scheme, which has been demonstrated to retain a much higher degree of fidelity than any of the data compression schemes using various wavelet bases. Song and Zhang (2001), Damerval et al. (2005) and Yuan et al. (2008) used a third way based on Delaunay Triangulation and on piecewise cubic polynomial interpretation to obtain an upper surface and a lower surface. Xu et al. (2006) provided the fourth approach by using a mesh fitting method based on finite elements. These BEMDs have accomplished some successes when they are applied to various fields of engineering and sciences.
Unfortunately, currently available genuine BEMDs, as those mentioned earlier, have several difficulties. The first one is the definition of extrema. All two-dimensional data have saddle, ridge and trough structures, one needs to make a decision of whether the saddle and ridge (trough) points should be considered maxima (minima). The fitting surfaces could be greatly different in the case of considering ridge (trough) points as maxima and in the case of not. Consequently, the decomposition results would be dramatically different. The second difficulty is that surface fitting can be computationally expensive. In many cases, it involves a very large matrix and its eigenvalue computations, and the fittings offer only an approximation and could not go through all the actual extrema. The third difficulty, and probably the most serious one, is the mode mixing, or more generally, scale mixing. In one-dimensional EMD, scale mixing is defined as a single IMF either consisting of signals of widely disparate scales, or a signal of a similar scale residing in different IMF components, caused by unevenly distributed extrema along the time axis (Huang et al. 1999, Wu and Huang 2005b, 2008). This difficulties of scale mixing can cause the similar problems in BEMD. For example, the decomposition of two-dimensional sinusoidal waves with part (suppose the lower-left quarter) of the data contaminated by noise could lead to the decomposed components of which none has the structure of the two-dimensional sinusoidal waves. Since two-dimensional data can contain noise, such decomposition using BEMD is unstable and can be very sensitive to noise, essentially leading to the physically un-interpretable results, similar to the one-dimensional case described by Huang et al. (1999) and Wu and Huang (2005b, 2008).